## Linear Operators: General theory |

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Page 57

A continuous

continuous

one-to- one

...

A continuous

**linear**one-to-one**map**of one F-space onto all of another has acontinuous

**linear**inverse. Proof. Let X, ^) be F-spaces and T a continuous**linear**one-to- one

**map**with TX = 2). Since (21-1)-1 = T**maps**open sets onto open sets...

Page 58

The

. Hence, by Theorem 2, its inverse pr^1 is continuous. Thus T = pr^pr^1 is

continuous (1.4.17). Q.E.D. 5 Theorem. // a

each of ...

The

**map**pr-g : [x, Tx]->x of & onto 3£ is one-to-one,**linear**, and continuous (1.8.3). Hence, by Theorem 2, its inverse pr^1 is continuous. Thus T = pr^pr^1 is

continuous (1.4.17). Q.E.D. 5 Theorem. // a

**linear**space is an F-space undereach of ...

Page 490

is in a space of continuous functions. In some other cases very little is known. For

example, while it is easy to see that the general continuous

1], p > 1, to L9[0, 1] has the form no satisfactory expression for the norm of T is ...

is in a space of continuous functions. In some other cases very little is known. For

example, while it is easy to see that the general continuous

**linear map**from Lp[0,1], p > 1, to L9[0, 1] has the form no satisfactory expression for the norm of T is ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero